National Australia Bank, parent company of National Irish Bank, recently announced free access to its Internet service for customers in Australia.
If the venture succeeds it is expected that a similar service will be unveiled in the Republic, said a spokeswoman.
The new service allows Australian customers to access the bank's website for the cost of a local telephone call.
The bank has also appointed an Internet service provider and will give customers links to 4,000 free websites and 75 hours of free access to pay sites every month.
In the Republic, National Irish Bank has a website, but falls short of providing Internet banking to its retail customers.
PC Banking is a free dial-up service available to business customers, allowing them to view their bank balances, transfer funds, payroll and send payments. Later in the year, NIB will finalise its TeleBank business and charges are yet to be announced.
Personal customers may avail of a similar transactional service next spring.